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Section 3 Local polynomial filters

In this section we detail the filters that arise from fitting a local polynomial to our time series, as described by Proietti and Luati (2008). Local polynomial filters encompass classical filters like Henderson and Musgrave filters (see sections 3.1.1 and 3.2.2).

We assume that our time series \(y_t\) can be decomposed as \[ y_t=\mu_t+\varepsilon_t \] where \(\mu_t\) is the signal (trend) and \(\varepsilon_{t}\overset{i.i.d}{\sim}\mathcal{N}(0,\sigma^{2})\) is the noise⁹. We assume that \(\mu_t\) can be locally approximated by a polynomial of degree \(d\) of the time \(t\) between \(y_t\) and the neighboring observations \(\left(y_{t+j}\right)_{j\in\left\llbracket -h,h\right\rrbracket}\). Then \(\mu_t\simeq m_{t}\) with: \[ \forall j\in\left\llbracket -h,h\right\rrbracket :\: y_{t+j}=m_{t+j}+\varepsilon_{t+j},\quad m_{t+j}=\sum_{i=0}^{d}\beta_{i}j^{i} \] This signal extraction problem is then equivalent to the estimation of \(m_t=\beta_0\). In matrix notation we can write: \[ \underbrace{\begin{pmatrix}y_{t-h}\\ y_{t-(h-1)}\\ \vdots\\ y_{t}\\ \vdots\\ y_{t+(h-1)}\\ y_{t+h} \end{pmatrix}}_{y}=\underbrace{\begin{pmatrix}1 & -h & h^{2} & \cdots & (-h)^{d}\\ 1 & -(h-1) & (h-1)^{2} & \cdots & (-(h-1))^{d}\\ \vdots & \vdots & \vdots & \cdots & \vdots\\ 1 & 0 & 0 & \cdots & 0\\ \vdots & \vdots & \vdots & \cdots & \vdots\\ 1 & h-1 & (h-1)^{2} & \cdots & (h-1)^{d}\\ 1 & h & h^{2} & \cdots & h^{d} \end{pmatrix}}_{X}\underbrace{\begin{pmatrix}\beta_{0}\\ \beta_{1}\\ \vdots\\ \vdots\\ \vdots\\ \vdots\\ \beta_{d} \end{pmatrix}}_{\beta}+\underbrace{\begin{pmatrix}\varepsilon_{t-h}\\ \varepsilon_{t-(h-1)}\\ \vdots\\ \varepsilon_{t}\\ \vdots\\ \varepsilon_{t+(h-1)}\\ \varepsilon_{t+h} \end{pmatrix}}_{\varepsilon} \] Two parameters are crucial in determining the accuracy of the approximation:

• the degree \(d\) of the polynomial;

• the number of neighbors \(H=2h+1\) (or the bandwidth \(h\)).

In order to estimate \(\beta\) we need \(H\geq d+1\) and the estimation is done by the weighted least squares (WLS), which consists of minimizing the following objective function: \[ S(\hat{\beta}_{0},\dots,\hat{\beta}_{d})=\sum_{j=-h}^{h}\kappa_{j}(y_{t+j}-\hat{\beta}_{0}-\hat{\beta}_{1}j-\dots-\hat{\beta}_{d}j^{d})^{2} \] where \(\kappa_j\) is a set of weights called kernel. We have \(\kappa_j\geq 0:\kappa_{-j}=\kappa_j\), and with \(K=diag(\kappa_{-h},\dots,\kappa_{h})\), the estimate of \(\beta\) can be written as \(\hat{\beta}=(X'KX)^{1}X'Ky\). With \(e_{1}=\begin{pmatrix}1&0&\cdots&0\end{pmatrix}'\), the estimate of the trend is: \[ \hat{m}_{t}=e_{1}\hat{\beta}=w'y=\sum_{j=-h}^{h}w_{j}y_{t-j}\text{ with }w=KX(X'KX)^{-1}e_{1} \] To conclude, the estimate of the trend \(\hat{m}_{t}\) can be obtained applying the symmetric filter \(w\) to \(y_t\)¹⁰. Moreover, \(X'w=e_{1}\) so: \[ \sum_{j=-h}^{h}w_{j}=1,\quad\forall r\in\left\llbracket 1,d\right\rrbracket :\sum_{j=-h}^{h}j^{r}w_{j}=0 \] Hence, the filter \(w\) preserve deterministic polynomial of order \(d\).

3.1 Different kernels

In signal extraction, observations are generally weighted according to their distance from time \(t\): this is the role of the kernel function. In the discrete case, a kernel function is a set of weights \(\kappa_j\), \(j=0,\pm1,\dots,\pm h\) with \(\kappa_j \geq0\) and \(\kappa_j=\kappa_{-j}\). An important class of kernels is the Beta kernels. In the discrete case, up to a proportional factor (so that \(\sum_{j=-h}^h\kappa_j=1\)): \[ \kappa_j = \left( 1- \left\lvert \frac j {h+1} \right\lvert^r \right)^s \] with \(r>0\), \(s\geq 0\). It encompasses all kernels used in this report, except Henderson, trapezoidal and gaussian kernel.The following kernels are considered in this report:

• \(r=1,s=0\) uniform kernel: \[\kappa_j^U=1\]

• \(r=s=1\) triangle kernel: \[\kappa_j^T=\left( 1- \left\lvert \frac j {h+1} \right\lvert \right)\]

• \(r=2,s=1\) Epanechnikov (or Parabolic) kernel: \[\kappa_j^E=\left( 1- \left\lvert \frac j {h+1} \right\lvert^2 \right)\]

• \(r=s=2\) biweight kernel: \[\kappa_j^{BW}=\left( 1- \left\lvert \frac j {h+1} \right\lvert^2 \right)^2\]

• \(r = 2, s = 3\) triweight kernel: \[\kappa_j^{TW}=\left( 1- \left\lvert \frac j {h+1} \right\lvert^2 \right)^3\]

• \(r = s = 3\) tricube kernel: \[\kappa_j^{TC}=\left( 1- \left\lvert \frac j {h+1} \right\lvert^3 \right)^3\]

• Henderson kernel (see section 3.1.1 for more details): \[ \kappa_{j}=\left[1-\frac{j^2}{(h+1)^2}\right] \left[1-\frac{j^2}{(h+2)^2}\right] \left[1-\frac{j^2}{(h+3)^2}\right] \]

• Trapezoidal kernel: \[ \kappa_j^{TP}= \begin{cases} \frac{1}{3(2h-1)} & \text{ if }j=\pm h \\ \frac{2}{3(2h-1)} & \text{ if }j=\pm (h-1)\\ \frac{1}{2h-1}& \text{ otherwise} \end{cases} \]

• Gaussian kernel¹¹: \[ \kappa_j^G=\exp\left( -\frac{ j^2 }{ 2\sigma^2h^2 }\right) \]

Henderson, trapezoidal and gaussian kernel are very specific:

• The Henderson and trapezoidal kernel functions change with the bandwidth (the other kernel only depend on the ratio \(j/h+1\)).

• Other definitions of the trapezoidal and gaussian kernel can be used. The trapezoidal kernel is here considered because it corresponds to the filter used to extract the seasonal component in the X-12ARIMA algorithm. Therefore it is never used to extract trend-cycle component.

Figure 3.1: Coefficients of the different kernels for \(h\) from 2 to 30.

The figure 3.1 summarizes the coefficients of the different kernels. Analyzing the coefficients we can already anticipate some properties of the associated filters:

• The triweight kernel has the narrowest distribution. The narrowest a distribution is, the smallest the weights of furthest neighbors are: the associated filter should have a high weight in the current observation (\(t\)).

• For \(h\) high the Henderson kernel is equivalent to the triweight kernel (since \(h+1\sim h+2 \sim h+3\), \(\kappa_j^H\sim\kappa_j^{TW}\)), the associated filter should also be equivalent. However, for \(h\) small (\(h\leq10\)) the Henderson kernel is closer to the biweight kernel than to the triweight kernel.

3.1.1 Specific symmetric filters

When \(p=0\) (local adjustment by a constant) we obtain the Nadaraya-Watson’s estimator.

With the uniform kernel we obtain the Macaulay filter. When \(p=0,1\), this is the arithmetic moving average: \(w_j=w=\frac{1}{2h+1}\).

The Epanechnikov kernel is often recommended as the optimal kernel that minimizes the mean square error of the estimation by local polynomial.

Loess is a locally weighted polynomial regression that uses tricube kernel.

The Henderson filter is a specific case of a local cubic fit (\(p=3\)), widely used for trend estimation (for example it’s the filter used in the seasonal adjustment software X-12ARIMA). For a fixed bandwidth, Henderson found the kernel that gave the smoothest estimates of the trend. He showed that the three following problems were equivalent:

1. minimize the variance of third difference of the series by the application of the moving average;
   
2. minimize the sum of squares of third difference of the coefficients of the filter, it’s the smoothness criterion: \(S=\sum_j(\nabla^{3}\theta_{j})^{2}\);
   
3. fit a local cubic polynomial by weighted least squares, where the weights are chose to minimize the sum of squares of the resulting filter.

Resolving the last problem leads to the kernel presented in section 3.1.

3.1.2 Analysis of symmetric filters

In this section, all the filters are computed by local polynomial of degree \(d=3\). The figure 3.2 plots the coefficients of the filters for the different kernels presented in different kernels presented in section 3.1 and for different bandwidths \(h\). The table 3.1 shows the variance reduction of the different filters. We find the similar results than in section 3.1:

• The triweight kernel gives the filter with the narrowest distribution. The narrowest a distribution is, the higher the variance reduction should be. Indeed, the distribution of the coefficients of the filter can be interpreted as the output signal of an additive outlier. As a result, with a wide distribution, an additive outlier will be more persistent than with a narrow distribution. Therefore, it’s the triweight that has the higher variance reduction for all \(h\leq30\).

• For \(h\) small, the trapezoidal filter seems to produce similar results than the Epanechnikov one.

• For \(h\) small the Henderson filter is closed to the biweight kernel, for \(h\) high it is equivalent to the triweight kernel.

Figure 3.2: Coefficients of symmetric filters computed by local polynomial of degree \(3\), according to the differents kernels and for \(h\) from 2 to 30.

Table 3.1: Variance reduction ratio (\(\sum\theta_i^2\)) of symmetric filters computed by local polynomial of degree \(3\).

	Kernel
\(h\)	Biweight	Epanechnikov	Gaussian	Henderson	Trapezoidal	Triangular	Tricube	Triweight	Uniform
2	0.50	0.49	0.49	0.50	0.51	0.51	0.49	0.52	0.49
3	0.33	0.30	0.30	0.32	0.31	0.33	0.32	0.37	0.28
4	0.25	0.23	0.24	0.25	0.23	0.25	0.25	0.28	0.22
5	0.22	0.21	0.21	0.22	0.20	0.22	0.22	0.24	0.20
6	0.20	0.19	0.19	0.20	0.19	0.20	0.20	0.21	0.19
7	0.19	0.18	0.18	0.19	0.18	0.19	0.19	0.20	0.18
8	0.18	0.17	0.18	0.18	0.17	0.18	0.18	0.19	0.17
9	0.17	0.17	0.17	0.17	0.17	0.17	0.17	0.18	0.17
10	0.17	0.16	0.17	0.17	0.16	0.17	0.17	0.17	0.16
20	0.12	0.12	0.13	0.13	0.12	0.12	0.13	0.13	0.12
30	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10

Moreover, we find that for all the filters, the coefficients decrease, when the distance to the central observation increases, until a negative value and then increase towards 0 (except for the uniform kernel). Negative coefficients might be disturbing but they arise from the cubic polynomial constraints. Indeed to preserve polynomial of degree 2 (and so 3) we need \(\sum_{j=-h}^hj^2\theta_i=0\), which constraint some coefficients to be negative. However, those negative coefficients are negligible compare to the central coefficients (they are more 80% smaller than the central coefficient for all kernels, except for uniform and trapezoidal with high bandwidth).

3.1.3 Gain functions

Figure 3.3 plots the gain functions of the different filters. Gain functions are usually plotted between 0 and \(\pi\). However, locally weighted polynomial regression are low-pass filters: they leave almost unchanged low frequency components (such as the trend) and attenuate high frequency fluctuations (noise). For a monthly data, a cycle of 3 years corresponds to the frequency \(2\pi/36\) and a cycle of 7 years to the frequency \(2\pi/84\). Therefore, an ideal pass-band filter will have a gain function equal to 1 for low frequency (\(\leq 2\pi/36\)) and equal to 0 for other frequencies.

When the bandwidth \(h\) increases, the gain function decreases for low frequencies: short business cycles will then be attenuated. For a fixed value of \(h\), gaussian, Henderson and triweight filters will preserve more short business cycles than the other filters (especially uniform, trapezoidal and Epanechnikov). Moreover, the gain function of those filters decreases faster to zero with less fluctuations: it enhances the higher variance reduction ratio shown in table 3.1.

Figure 3.3: Gain functions from 0 to \(2\pi/12\) of symmetric filters computed by local polynomial of degree \(3\), according to the differents kernels and for \(h\) from 2 to 30.

Note: the two horizontal lines corresponds to the frequencies \(2\pi/84\) (cycle of 7 years) and \(2\pi/36\) (cycle of 3 years).

Just analyzing the symmetric filters properties, there is no doubt that Henderson, triweight and biweight filters have similar properties and will perform better than the other kernel for trend-cycle extraction. The same results are found with asymmetric filters. Thus, in order to simplify the presentation analysis, in the next sections we will only show the results with the Henderson filter.

3.2 Asymmetric filters

3.2.1 Direct asymmetric filters (DAF)

As mentioned in section 2.3, symmetric filters cannot be used in boundary points. For real-time estimation, three different approaches can be used:

1. Build an asymmetric filter fitting local polynomial to the available observations \(y_{t}\) for \(t\in\left\llbracket n-h,n\right\rrbracket\).

2. Apply the symmetric filter to the series extended by forecast (or backcast) \(\hat{y}_{n+l\mid n},l\in\left\llbracket 1,h\right\rrbracket\).

3. Build an asymmetric filter which minimize the mean square revision error subject to polynomial reproducing constraints.

Proietti and Luati (2008) show that the first two approaches are equivalent when the forecast is done by a polynomial extrapolation of order \(d\) (forecasts generated with the same polynomial model than the symmetric filter). This is called the direct asymmetric filter (DAF). Let \(q\) be the number of available observations in the future: \(q\) varies from 0 (real time filter) to \(h\) (symmetric filter).

Rewriting the matrix \(X\), \(K\) \(y\) in the following way: \[ X=\begin{pmatrix}X_{p}\\ X_{f} \end{pmatrix},\quad y=\begin{pmatrix}y_{p}\\ y_{f} \end{pmatrix},\quad K=\begin{pmatrix}K_{p} & 0\\ 0 & K_{f} \end{pmatrix} \] where \(y_{p}\) correspond to the available data and \(y_{f}\) the missing data. The DAF \(w_a\) and the forecast \(\hat{y}_{f}\) can be written as: \[ w_{a}=K_{p}X_{p}(X'_{p}K_{p}X_{p})^{-1}e_{1}, \quad \hat{y}_{f}=X_{f}(X'_{p}K_{p}X_{p})^{-1}X_{p}'K_{p}y_{p} \] Moreover, we have the following results with the DAF \(w_a\):

• it satisfies the same polynomial reproduction constraints as the symmetric filter (conserve polynomial of degree \(d\)). Thus, the bias in estimating an unknown function of time has the same order of magnitude as in the interior of time support.

• \(w_a\) minimize the weighted distance (by the kernel function) between the asymmetric filter coefficients and the symmetric ones. Therefore, for the DAF it is equivalent to fit a local polynomial and to minimize the revisions

However, the weights \(w_{a,0}\) of the DAF are highly concentrated in the current observation \(t\) with an important change between \(q=0\) (real-time filter) and \(q=h\) (see figure 3.4). Moreover the real-time filter doesn’t have a satisfying gain functions: it is closer to one for all the frequencies (it thus has a low noise reduction power). Therefore, even if the real-time filter is unbiased (if the series is generated by a polynomial of degree \(d\)) it is at the expenses of a high variance.

Figure 3.4: Coefficients and gain function of direct asymmetric filters (DAF) computed by local polynomial of degree \(3\) with the Henderson kernel for \(h=6\).

For all the kernels, we find the same results as in Proietti and Luati (2008):

• For a fixed value of \(d\), the more the data is available (\(q\) increases), the more the weight associated to the current observation \(w_{a,0}\) decreases.

• For a fixed value of \(h\) and \(q\), \(w_{a,0}\) increases exponentially with the polynomial degree \(d\) (in particular, for \(d=h\), \(w_{a,0}=1\)).

3.2.2 General class of asymmetric filters

To deal with the problem of the variance of the estimates of the real-time filters, Proietti and Luati (2008) suggest a general of asymmetric filters to make a tradeoff between bias and variance. It is a generalisation of Musgrave asymmetric filters (used in the seasonal adjustment algorithm X-12ARIMA, see Musgrave (1964)).

Here we consider that the data is generated by the model: \[ y=U\gamma+Z\delta+\varepsilon,\quad \varepsilon\sim\mathcal{N}(0,D) \] The goal is to find a filter \(v\) which minimize the mean square revision error (with the symmetric filter \(w\)) subject to some constraints. The constraints are summarized by the matrix \(U=\begin{pmatrix}U_{p}'&U_{f}'\end{pmatrix}'\) (with \(U_p\) the available observations of the matrix \(U\) for the asymmetric filter): \(U_p'v=U'w\). The problem is equivalent to find \(v\) that minimize: \[\begin{equation} \varphi(v)= \underbrace{ \underbrace{(v-w_{p})'D_{p}(v-w_{p})+ w_{f}'D_{f}w_{f}}_\text{revision error variance}+ \underbrace{[\delta'(Z_{p}'v-Z'w)]^{2}}_{biais^2} }_\text{Mean square revision error}+ \underbrace{2l'(U_{p}'v-U'w)}_{\text{constraints}} \tag{3.1} \end{equation}\] with \(l\) a vector of Lagrange multipliers.

When \(U=X\) this is equivalent to the constraint to preserve polynomial of degree \(d\): we find the direct asymmetric filters \(w_a\) with \(D=K^{-1}\).

When \(U=\begin{pmatrix}1&\cdots&1\end{pmatrix}'\), \(Z=\begin{pmatrix}-h&\cdots&+h\end{pmatrix}'\), \(\delta=\delta_1\), \(D=\sigma^2I\) and when the symmetric filter is the Henderson filter we obtain the Musgrave asymmetric filters. With this filter we assume that the data is generated by a linear process and that the asymmetric filters preserve constant signals (\(\sum v_i=\sum w_i=1\)). The asymmetric filters depends on the ratio \(\delta_1/\sigma\), which is related to the “I-C” ratio \(R=\frac{\bar{I}}{\bar{C}}=\frac{\sum\lvert I_t-I_{t-1}\rvert}{\sum\lvert C_t-C_{t-1}\rvert}\) (\(\delta_1/\sigma=2/(R\sqrt{\pi})\)), the ratio between the expected absolute difference of the irregular and of the trend-cycle. In the seasonal adjustment method, the I-C ratio¹² is used to determine the bandwidth to use for the Henderson filter. For monthly data:

• if \(R<1\) a 9-term Henderson is used (\(h=4\));

• if \(1\leq R\leq3.5\) a 13-term Henderson is used (\(h=6\));

• if \(3.5< R\) a 23-term Henderson is used (\(h=12\)).

In this report, for simplicity we only consider 13-term symmetric filters: the ratio \(\delta^2/\sigma^2\) is fixed to \(3.5\).

When \(U\) corresponds to the first \(d^*+1\) first the columns of \(X\), \(d^*<d\), the constraint is that the asymmetric filter should reproduce polynomial of degree \(d^*\), the potential bias depends on the value of \(\delta\). This will reduce the variance at the expense of a bias: it is the idea followed by Proietti and Luati (2008) to propose three class of asymmetric filters:

1. Linear-Constant (LC): \(y_t\) linear (\(d=1\)) and \(v\) preserves constant signals (\(d^*=0\)). We obtain Musgrave filters when the Henderson kernel is used.

2. Quadratic-Linear (QL): \(y_t\) quadratic (\(d=2\)) and \(v\) preserves linear signals (\(d^*=1\)).

3. Cubic-Quadratic (CQ): \(y_t\) cubic (\(d=3\)) and \(v\) preserves quadratic signals (\(d^*=2\)).

The table 3.2 shows the quality criteria of the different methods with the Henderson kernel and \(h=6\). For real-time filters (\(q=0\)), the more complex the filter is (in terms of polynomial preservation), the less the timeliness is and the more the fidelity/smoothness is: the reduction of the time-delay is at the expense of an increased variance. This change when \(q\) increases: for \(q=2\) the QL filter has a greater timeliness that the LC filter. This unexpected result underlines the fact that in the approach of Proietti and Luati (2008), the timeliness is never set as a goal to minimize.

Table 3.2: Quality criteria of asymmetric filters (\(q=0,1,2\)) computed by local polynomial with Henderson kernel for \(h=6\) and \(R=3.5\).

Method	\(b_c\)	\(b_l\)	\(b_q\)	\(F_g\)	\(S_g\)	\(T_g \times 10^{-3}\)	\(A_w\)	\(S_w\)	\(T_w\)	\(R_w\)
\(q=0\)
LC	0	-0.407	-2.161	0.388	1.272	30.341	0.098	0.488	0.409	0.548
QL	0	0.000	-0.473	0.711	5.149	0.047	0.067	1.894	0.000	0.106
CQ	0	0.000	0.000	0.913	11.942	0.015	0.016	2.231	0.000	0.102
DAF	0	0.000	0.000	0.943	14.203	0.003	0.015	2.178	0.000	0.098
\(q=1\)
LC	0	-0.121	-0.525	0.268	0.433	4.797	0.009	0.119	0.063	0.112
QL	0	0.000	-0.061	0.287	0.707	0.694	0.005	0.192	0.007	0.042
CQ	0	0.000	0.000	0.372	0.571	0.158	0.022	0.575	0.001	0.061
DAF	0	0.000	0.000	0.409	0.366	0.061	0.020	0.760	0.000	0.059
\(q=2\)
LC	0	0.003	1.076	0.201	0.080	0.347	0.009	0.012	0.004	0.015
QL	0	0.000	0.033	0.215	0.052	2.083	0.000	0.011	0.023	0.067
CQ	0	0.000	0.000	0.370	0.658	0.131	0.021	0.558	0.001	0.055
DAF	0	0.000	0.000	0.398	0.768	0.023	0.017	0.677	0.000	0.048

Regarding the mean square revision error (\(A_w+S_w+T_w+R_w\)), LC and QL filters always gives better results than CQ and DAF filters. This “theorical” mean square revision error can be compared to the “empirical” one computed applying the filters to real data. The table 3.3 shows the average mean square revision error of the different filters applied to the Industrial production indices of the European Union between 2003 and 2019. The mean square revision errors are different in level but give same results comparing the different filters: this validate the formula used for the criteria \(A_w,S_w,T_w,\text{ and }R_w\) (see section 5).

Taking for the value \(R\) the “I-C” ratio computed with the raw data gives the same summary results.

Table 3.3: Mean squared revision error of asymmetric filters (\(q=0,1,2\)) computed by local polynomial on the Industrial production indices of the European Union.

	Mean squared revision error
Method	2007-2010	2003-2019	\(A_w+S_w+T_w+R_w\)
\(q=0\)
LC	1.54	0.86	1.54
QL	2.04	1.56	2.07
CQ	2.88	2.36	2.35
DAF	3.03	2.49	2.29
\(q=1\)
LC	2.96	1.32	0.30
QL	2.31	1.07	0.25
CQ	2.62	1.34	0.66
DAF	2.72	1.44	0.84
\(q=2\)
LC	6.78	2.77	0.04
QL	7.45	3.07	0.10
CQ	7.99	3.62	0.64
DAF	7.94	3.62	0.74
Note: the filters are computed with \(h=6\) (13-terms symmetric filter) and \(R=3.5\).

All those results suggest focusing on LC and QL filters and to focus on asymmetric linear filters that preserve polynomial trends of degree equal or less than one.

The results for the different kernels can also be visualized in an online application available at https://aqlt.shinyapps.io/FiltersProperties/.

Local polynomial filters

Advantages:

• Simple models with an easy interpretation.

• The asymmetric linear filter is independent of the date of estimation. However, it depends on the data if we calibrate the filter with the “I-C” ratio.

Drawbacks:

• Timeliness is not controlled.

References

Musgrave, John C. 1964. “A Set of End Weights to End All End Weights.” US Census Bureau [Custodian]. https://www.census.gov/ts/papers/Musgrave1964a.pdf.

Proietti, Tommaso, and Alessandra Luati. 2008. “Real Time Estimation in Local Polynomial Regression, with Application to Trend-Cycle Analysis.” Ann. Appl. Stat. 2 (4): 1523–53. https://doi.org/10.1214/08-AOAS195.

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9. The time series is therefore seasonally adjusted.↩︎

10. \(w\) is symmetric due to the symmetry of the kernel weights \(\kappa_j\).↩︎

11. In this report we arbirarily take \(\sigma^2=0.25\).↩︎

12. To compute the I-C ratio, a first decomposition of the seasonally adjusted series is computed using a 13-term Henderson moving average.↩︎